One thing to note when relating symmetric group with affine braid group is that?

every braid on n strands determines a permutation on n elements

every braid on n strands are different on n elements

every braid on n strands determines a permutation on p elements

every braid on n strands is linear to n elements

Do You Know Affine Braid Group?

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Questions: 10 | Attempts: 118 | Updated: Mar 19, 2023

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What is your knowledge on Affine Braid Group? You can find out by taking this quiz. On a brief note, An Affine Braid Group is a braid group associated to an affine Coxeter system. Their group rings have quotients called affine Hecke algebras. They are subgroups of double affine braid groups. How well do you know about this? Choose the correct answers to the accompany questions to find out.

Questions and Answers

1.

The generalization to other values of n will be straightforward in an affine braid group, if?
- A.
  
  N=4
- B.
  
  N=2
- C.
  
  N=0
- D.
  
  N=-2
Correct Answer
A. N=4

Explanation
In an affine braid group, the generalization to other values of n will be straightforward if N=4. This means that when N is equal to 4, it will be easy to extend the generalization to include other values of n in the affine braid group. However, without further context or information about the specific properties of the affine braid group, it is difficult to provide a more detailed explanation.

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2.

In an affine braid group the set of all braids on four strands is denoted by?
- A.
  
  B4
- B.
  
  B3
- C.
  
  B2
- D.
  
  B7
Correct Answer
A. B4

Explanation
In an affine braid group, the set of all braids on four strands is denoted by B4. This notation is commonly used to represent the group of braids on four strands in the context of affine braid groups.

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3.

One major property of affine braid group is that all non-identity elements of Bn posses?
- A.
  
  Infinite order
- B.
  
  Linear order
- C.
  
  Finite order
- D.
  
  Parallel order
Correct Answer
A. Infinite order

Explanation
The correct answer is "Infinite order". This means that all non-identity elements of the affine braid group have an infinite number of repetitions before they return to their original position. In other words, they do not have a finite period or cycle. This property distinguishes the affine braid group from other groups where elements may have a finite order or cycle.

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4.

The left-invariant linear order on Bn in a braid group is known as?
- A.
  
  Infinite order
- B.
  
  Dehomoy order
- C.
  
  Parallel order
- D.
  
  Finite prism
Correct Answer
B. Dehomoy order

Explanation
The left-invariant linear order on Bn in a braid group is known as the Dehomoy order. This order is defined by comparing the positions of the strands in a braid, where a braid with lower positions is considered smaller in the Dehomoy order. It is called "left-invariant" because the order remains the same when multiplying a braid by another braid on the left. This order is important in the study of braid groups and their properties.

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5.

One thing to note when relating symmetric group with affine braid group is that?
- A.
  
  Every braid on n strands are different on n elements
- B.
  
  Every braid on n strands determines a permutation on p elements
- C.
  
  Every braid on n strands is linear to n elements
- D.
  
  Every braid on n strands determines a permutation on n elements
Correct Answer
D. Every braid on n strands determines a permutation on n elements

Explanation
The correct answer is that every braid on n strands determines a permutation on n elements. This means that for every braid in the symmetric group, there is a corresponding permutation on the same number of elements. The braid group and the symmetric group are closely related, and this relationship allows us to associate permutations with braids and vice versa.

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6.

Which of the following braid group is the universal central extension of the modular group PSL(2, Z)?
- A.
  
  B1
- B.
  
  B2
- C.
  
  B3
- D.
  
  B4
Correct Answer
C. B3

Explanation
The braid group B3 is the universal central extension of the modular group PSL(2, Z). This means that B3 is a larger group that contains PSL(2, Z) as a normal subgroup, and every central extension of PSL(2, Z) is isomorphic to B3.

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7.

The affine group that posses group rings with quotients called double affine Hecke algebras is called.
- A.
  
  Double affine group
- B.
  
  Hecke affine group
- C.
  
  Ring based affine group
- D.
  
  Weyl affine group
Correct Answer
A. Double affine group

Explanation
The correct answer is Double affine group. The affine group that possesses group rings with quotients called double affine Hecke algebras is known as the Double affine group. This group is a generalization of the affine Weyl group and has applications in various areas of mathematics, such as algebraic geometry and representation theory.

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8.

An affine braid group is a braid group generally associated with which mathematical system?
- A.
  
  Affine ring system
- B.
  
  Affine Coxeter system
- C.
  
  Affine Coter system
- D.
  
  Affine double system
Correct Answer
B. Affine Coxeter system

Explanation
An affine braid group is a braid group generally associated with an affine Coxeter system. An affine Coxeter system is a mathematical system that consists of a set of generators and relations, which are used to define a group. In the case of an affine braid group, the generators and relations are specifically chosen to represent braids. Therefore, the correct answer is affine Coxeter system.

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9.

One way to represent concretely the Elements of the braid group is via?
- A.
  
  Decimals
- B.
  
  Indices
- C.
  
  Matrices
- D.
  
  Coefficients
Correct Answer
C. Matrices

Explanation
Matrices can be used to represent the elements of the braid group concretely. The braid group is a mathematical concept that involves the study of braids and their properties. Matrices provide a convenient and efficient way to represent the elements of the braid group because they can capture the necessary information and operations involved in manipulating the braids. By using matrices, it becomes easier to perform calculations and analyze the properties of the braid group.

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10.

The Lawrence representations depending on several parameters for affine brai group was postulated in what year?
- A.
  
  1990
- B.
  
  1992
- C.
  
  1991
- D.
  
  1993
Correct Answer
A. 1990

Explanation
The Lawrence representations for affine brai group were postulated in 1990.

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Do You Know Affine Braid Group?

The generalization to other values of n will be straightforward in an affine braid group, if?

In an affine braid group the set of all braids on four strands is denoted by?

One major property of affine braid group is that all non-identity elements of Bn posses?

The left-invariant linear order on Bn in a braid group is known as?

One thing to note when relating symmetric group with affine braid group is that?

Which of the following braid group is the universal central extension of the modular group PSL(2, Z)?

The affine group that posses group rings with quotients called double affine Hecke algebras is called.

An affine braid group is a braid group generally associated with which mathematical system?

One way to represent concretely the Elements of the braid group is via?

The Lawrence representations depending on several parameters for affine brai group was postulated in what year?